Mathematics 241, section 1 – Multivariable Calculus Exam III – November 26, 2013

Directions

Do all work in the blue exam booklet. There are 100 regular points and 10 Extra Credit points.

I. (20) Given that

f(x, y) = \frac{y}{x^{2} + y^{2} + 4},

and

\begin{matrix}f_{x} = \frac{−2xy}{(x^{2} + y^{2} + 4)^{2}} & f_{y} = \frac{x^{2} − y^{2} + 4}{(x^{2} + y^{2} + 4)^{2}} \\ f_{xx} = \frac{2y(3x^{2} − y^{2} − 4)}{(x^{2} + y^{2} + 4)^{3}} f_{xy} & = \frac{−2x(x^{2} − 3y^{2} + 4)}{(x^{2} + y^{2} + 4)^{3}} f_{yy} = \frac{−2y(3x^{2} − y^{2} + 12)}{(x^{2} + y^{2} + 4)^{3}} \end{matrix}

find all critical points of f and apply the Second Derivative Test to each to determine the types of the critical points.

II. A) (20) Find the maximum and minimum values of

f(x, y) = xy + 2y

on the closed and bounded set

S = {(x, y) ∈ R^{2} | x^{2} + y^{2} ≤ 9}.

B) (5) Give a sketch of the contours of f for the maximum and minimum values you found in part A, together with S.

III. All parts of this problem refer to the region

R = {(x, y) ∈ R^{2} | x^{2} + y^{2} ≤ 4, y ≥ 0, y ≤ 2x}.

and the solid S ⊂ R^{3} with upper boundary z = x^{2} and lower boundary the xy-plane.

A) (10) Sketch the region R in the plane. B) (10) Set up iterated integral(s) of the form ∫ ∫ f(x, y) dy dx to compute the volume

of S. Do not compute the value. C) (10) Now set up iterated integral(s) of the form ∫ ∫ f(x, y) dx dy to compute the

volume of S. Do not compute the value. D) (10) Finally, set up a polar coordinate integral that will compute the volume of S. Do

not compute the value. E) (15) Finally, compute the volume of S using any one of your answers from parts B,C,D

(your choice). For full credit, you must show all work leading to your final answer.

Extra Credit (10) Evaluate the improper double integral

∫ ∫_{R^{2}} e^{1−x^{2}−y^{2}} dA = ∫_{−∞}^{∞} ∫_{−∞}^{∞} e^{1−x^{2}−y^{2}} dy dx.